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Electronic Journal of Mathematical Analysis and Applications

Vol. 1(1) Jan. 2013, pp. 47-54.

ISSN: 2090-792X (online)

http://ejmaa.6te.net/

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DISCONTINUOUS DYNAMICAL SYSTEMS GENERATED BY A

SEMI-DISCRETIZATION PROCESS

A. M. A. EL-SAYED, S. M. SALMAN

Abstract. In this paper, a semi-discretization process is applied to the well-known

Logistic differential equation. The new system after discretization will be discon-tinuous. Stability of fixed points for the resultant discontinuous system is studied.

Bifurcation analysis and chaos are discussed.

1. Introduction

A new class of discontinuous dynamical systems generated by retarded functional equa-tions has been defined in [3]-[7]. Here we are concerned with the well-known Logisticequation given by

x′(t) = ρx(t)(1− x(t)), t ∈ (0, T ],

with the initial condition

x(0) = xo,

where ρ is a positive constant. This equation has been studied and widely discussed inmany papers and textbooks of dynamical systems. In this work, we are interested in thediscontinuous counterpart of this equation which we are going to define and discuss itsbifurcation and chaos.

2. Discontinuous dynamical systems

Consider the problem of retarded functional equation

x(t) = f(x(t− r)), t ∈ (0, T ], (2.1)


x(τ) = φ(τ), τ ≤ 0. (2.2)

2000 Mathematics Subject Classification. 2010 MSC:39B05, 37N30, 37N20.Key words and phrases. Semi-discretization process, Logistic differential equation, discontinuous dy-

namical systems , fixed points, stability, bifurcation, chaos.Submitted Sep. 30, 2012. Published Jan 1, 2013.

47

48 A. M. A. EL-SAYED, S. M. SALMAN EJMAA-2013/1

If T is a positive integer, r = 1, φ(0) = xo and t = n = 1, 2, 3, ..., then the problem(2.1)-(2.2) will be the discrete dynamical system

xn = f(n, xn−1), n = 1, 2, 3, ..., T (2.3)

x(0) = xo. (2.4)

This shows that the discrete dynamical system (2.3)-(2.4) is a special case of the problemof the retarded functional equation (2.1)-(2.2).Let t ∈ (0, r], then t− r ∈ (−r, 0] and the solution of (2.1)-(2.2) is given by

x(t) = x1(t) = f(φ(0)), t ∈ (0, r].

For t ∈ (r, 2r], then t− r ∈ (0, r] and the solution of (2.1) is given by

x(t) = x2(t) = f(x1(t)) = f(f(φ(0))) = f2(φ(0)), t ∈ (r, 2r].

Repeating the process we can easily deduce that the solution of (2.1) is given by

x(t) = xn(t) = fn(φ(0)), t ∈ ((n− 1)r, nr],

which is continuous on each subinterval ((k − 1)r, kr] , k = 1, 2, 3, ..., n, but

limt→kr+

x(k+1)r(t) = fk+1(φ(0)) 6= xkr,

which implies that the solution of the problem (2.1)-(2.2) is discontinuous (sectionallycontinuous) on (0, T ] and thus we have proved the following theorem [3].

Theorem 1. The solution of the problem of retarded functional equation (2.1)-(2.2) isdiscontinuous (sectionally continuous) even if the functions f and φ are continuous.

Now let f : [0, T ]× Rn+1 → R and r1, r2, ..., rn ∈ R+. Then, the following definition canbe given

Definition 1. The discontinuous dynamical system is the problem of retarded functionalequation

x(t) = f(t, x(t− r1), x(t− r2), ..., x(t− rn)), t ∈ (0, T ], (2.5)

x(τ) = φ(τ), τ ≤ 0. (2.6)

Definition 2. The fixed points of the discontinuous dynamical system (2.5)-(2.6) are thesolution of the equation

x(t) = f(t, x, x, ..., x). (2.7)

3. A semi-discretization process

Consider the Logistic differential equation given by

x′(t) = ρx(t)(1− x(t)), t ∈ (0, T ], (3.1)

with the initial conditionx(0) = xo.

Now we are going to apply a semi-discretization process to this equation in order to obtainits discontinuous counterpart.Let r > 0 be given. Using The approximations

x′(t) ' x(t+ r)− x(t)

r,

EJMAA-2013/1DISCONTINUOUS DYNAMICAL SYSTEMS GENERATED BY A SEMI-DISCRETIZATION PROCESS49

in (3.1) we get

x(t+ r)− x(t)

r' ρx(t)(1− x(t)),

x(t+ r) = x(t) + ρx(t)(1− x(t)),

x(t) = x(t− r) + ρx(t− r)(1− x(t− r)).

That is, we have the following discontinuous counterpart of the Logistic equation whichis given by

x(t) = x(t− r) + rρx(t− r)(1− x(t− r)), t ∈ (0, T ], (3.2)


x(t) = x0, t ≤ 0.

By the same way as in section 2, we can show that the solution of system (3.2) fort ∈ (nr, (n+ 1)r] is given by the loop

xn+1(t) = xn(nr) + ρrxn(nr)(1− xn(nr)), n = 1, 2, 3, ...

It is worth to mention here that the previous semi-discretization process can be obtainedby Taylor expansion as follows:

x(t) = x(t− r) + x′(t− r) t− (t− r)1!

+ ...

' x(t− r) + rρx(t− r)(1− x(t− r)).

To summarize, we showed that there exists a semi-discretization process which generatea discontinuous dynamical system. This discontinuous dynamical system generalizes thediscrete one studied in [1].Figure (1) shows the trajectory of the discontinuous system (3.2) when r = 1, 0.5 and0.25, while Figure (2) shows the solution of the continuous system (3.1).

Figure 1. Trajectory of(3.2) with ρ = 0.5.

Figure 2. Solution of (3.1)with ρ = 0.5.


3.1. Approximate Solution. In this part we show that the proposed semi-discretizationprocess best approximates the solution of the discontinuous dynamical system (3.2) tothe exact solution of the continuous system (3.1) as shown in the table below. Now fort = n+n+1

2 r, the following table gives the absolute error = |exact−approximate| for somedifferent values of n and r. Figures (3)-(6) illustrate the approximate solution for (3.2)

n r=0.1 r=0.2 r=0.3 r=0.4 r=1

10 0.0656 0.1382 0.2019 0.2443 0.133820 0.0697 0.1123 0.0968 0.0591 1.0000e-00430 0.0647 0.0583 0.0231 0.0064 040 0.0529 0.0229 0.0043 6.0000e-004 050 0.0389 0.0079 7.0000e-004 0 0

Table 1. Absolute error

when r = 0.1, 0.2, 0.3,0.4 and 1, respectively.

Figure 3. Exact(solid)of (3.1) and approxi-mate(dashed) of (3.2) whenr = 0.1.





4. Fixed points and stability

The discontinuous dynamical system (3.2) has two fixed points namely, x1 = 0 and x2 = 1which can be easily obtained by solving the equation

x = x+ rρx(1− x).

To study the stability of these fixed points we need the following theorem

Theorem 2. [2] Let f be a smooth map on R, and assume that x0 is a fixed point of f.

1. If |f ′(x0)| < 1, then x0 is stable.

2. If |f ′(x0)| > 1, then x0 is unstable.

In our case, f = x = x+ rρx(1− x). Its derivative is f ′ = 1 + ρr− 2ρrx. That is, x1 = 0is stable if |1 + rρ| < 1, which is impossible, this means that x1 is unstable.Similarly, the second fixed point x2 = 1 is stable if |1− rρ| < 1.

5. Bifurcation and chaos

In this section we show by numerical experiments illustrated by bifurcation diagrams thatthe dynamical behavior of the discontinuous system (3.2) is completely affected by thechange in both r and T .Take r = 1 and t ∈ [0, 150] in (3.2) (Figure (7)).Take r = 0.1 and t ∈ [0, 15] in (3.2) (Figure (8)).Take r = 0.25 and t ∈ [0, 3] in (3.2) (Figure (9)).Take r = 0.5 and t ∈ [0, 6] in (3.2) (Figure (10)).Take r = 0.3 and t ∈ [0, 3] in (3.2) (Figure (11)).


Take r = 0.1 and t ∈ [0, 2] in (3.2) (Figure (12)).

Figure 7. Bifurcation dia-gram of (3.2) when r = 1 andt ∈ [0, 150].

Figure 8. Bifurcation dia-gram of (3.2) when r = 0.1and t ∈ [0, 15].






6. Conclusion

Applying a semi-descretization process to the well-known Logistic equation resulting in adiscontinuous dynamical system representing the Logistic map. Moreover, changing boththe retardation parameter r together with the time t ∈ [0, T ], has the strong effect onboth bifurcation and chaos behavior of the map.Figures (11) and (12) agree with our results, while Figures (7)-(8) and (9)-(10) indicatethat there is a scale which gives identical chaotic behavior.

References

[1] Benito Chen , Francisco Solis, Discretizations of nonlinear differential equations using explicitfinite order methods, Journal of Computational and Applied Mathematics 90 (1998) 171-183.

[2] S. N. Elaydi, An Introduction to Difference Equations, Third Edition, Undergraduate Textsin Mathematics, Springer, New York, 2005.

[3] A. M. A. El-Sayed, A. El-Mesiry and H. EL-Saka, On the fractional-order logistic equation,Applied Mathematics Letters, 20, 817-823, (2007).

[4] A. M. A. El-Sayed and M. E. Nasr, Existence of uniformly stable solutions of nonau-tonomous discontinuous dynamical systems J. Egypt Math. Soc. Vol. 19(1) (2011).

[5] A. M. A. El-Sayed and M. E. Nasr, On some dynamical properties of discontinuous dynamicalsystems, American Academic and Scholarly Research Journal Vol.2, No.1 (2012) pp. 28-32.

[6] A. M. A. El-Sayed and M. E. Nasr, On some dynamical properties of the discontinuousdynamical system represents the Logistic equation with different delays, I-manager’s Journalon Mathematics ’accepted’.

[7] A.M.A. El-Sayed and S. M. Salman, Chaos and bifurcation of discontinuous dynamical sys-tems with piecewise constant arguments, Malaya Journal of Matematik 1(1)(2012) 1519.

[8] John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.[9] Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

[10] Z. J. Jing and J. Yang, Chaos Solitons Fractals 27, 259 (2006).[11] X. Liu and D. Xiao, Chaos Solitons Fractals 32, 80 (2007).


[12] Kuznetsov Yu.A. (2004) Elements of Applied Bifurcation Theory, Springer, 3rd edition.

AcknowledgementThe authors thank Prof. Dr. E. Ahmed for his valuable comments.

Ahmed M. A. El-SayedFaculty of Science, Alexandria University, Alexandria, Egypt

E-mail address: [email protected], [email protected]

S. M. Salman

Faculty of Education, Alexandria University, Alexandria, Egypt

E-mail address: [email protected]

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